Euclidean Distance is defined as the distance between two points in Euclidean space. To find the distance between two points, the length of the line segment that connects the two points should be measured.
In this article, we will explore what is Euclidean distance, the Euclidean distance formula, its derivation, examples, etc.
What is Euclidean Distance?
The measure which gives the distance between any two points in an n-dimensional plane is known as Euclidean Distance. Euclidean distance between two points in the Euclidean space is defined as the length of the line segment between two points. It has various applications in mathematics.
Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space; the Euclidean Distance between them is given by using the formula:
d = √[(x2 – x1)2 + (y2 – y1)2]
where,
- d is Euclidean Distance
- (x1, y1) is Coordinate of the first point
- (x2, y2) is Coordinate of the second point
If the two points (x1, y1, z1) and (x2, y2, z2) are in a 3-dimensional space, the Euclidean Distance between them is given by using the formula:
d = √[(x2 – x1)2 + (y2 – y1)2+ (z2 – z1)2]
where,
- d is Euclidean Distance
- (x1, y1, z1) is Coordinate of the first point
- (x2, y2, z2) is Coordinate of the second point
In general, the Euclidean Distance formula between two points (x11, x12, x13, …., x1n) and (x21, x22, x23, …., x2n) in an n-dimensional space is given by the formula:
d = √[∑(x2i – x1i)2]
where,
- i Ranges from 1 to n
- d is Euclidean distance
- (x11, x12, x13, …., x1n) is Coordinate of First Point
- (x21, x22, x23, …., x2n) is Coordinate of Second Point
Euclidean Distance Formula is derived by following the steps added below:
Step 1: Let us consider two points, A (x1, y1) and B (x2, y2), and d is the distance between the two points.
Step 2: Join the points using a straight line (AB).
Step 3: Now, let us construct a right-angled triangle whose hypotenuse is AB, as shown in the figure below.
Step4: Now, using Pythagoras theorem we know that,
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
=> d2 = (x2 – x1)2 + (y2 – y1)2
Now, take the square root on both sides of the equation, we get
d = √(x2 – x1)2 + (y2 – y1)2
Solved Questions on Euclidean Distance
Here are some sample problems based on the distance formula.
Question 1: Calculate the distance between the points (4,1) and (3,0).
Solution:
Using Euclidean Distance Formula:
=> d = √(x2 – x1)2 + (y2 – y1)2
=> d = √(3 – 4)2 + (0 – 1)2
=> d = √(1 + 1)
=> d = √2 = 1.414 unit
Question 2: Show that the points A (0, 0), B (4, 0), and C (2, 2√3) are the vertices of an Equilateral Triangle.
Solution:
To prove that these three points form an equilateral triangle, we need to show that the distances between all pairs of points, i.e., AB, BC, and CA, are equal.
Distance between points A and B:
AB = √[(4– 0)2 + (0-0)2]
=> AB = √16
AB = 4 unit
Distance between points B and C:
BC = √[(2-4)^2 + (2√3-0)^2]
=> BC = √[4+12] = √16
BC = 4 unit
Distance between points C and A:
CA = √[(0-2)2 + (0-2√3)2]
=> CA = √[4 + 12] = √16
CA = 4 unit
Here, we can observe that all three distances, AB, BC, and CA, are equal.
Therefore, the given triangle is an Equilateral Triangle
Question 3: Mathematically prove Euclidean distance is a non negative value.
Solution:
Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space; the Euclidean Distance between them is given by using the formula:
d = √(x2 – x1)2 + (y2 – y1)2
We know that squares of real numbers are always non-negative.
=>(x2 – x1)2 >= 0 and (y2 – y1)2 >= 0
=> √(x2 – x1)2 + (y2 – y1)2 >= 0
As square root of a non-negative number gives a non-negative number,
Therefore Euclidean distance is a non-negative value. It cannot be a negative number.
Question 4: A triangle has vertices at points A(2, 3), B(5, 7), and C(8, 1). Find the length of the longest side of the triangle.
Solution:
Given, the points A(2, 3), B(5, 7), and C(8, 1) are the vertices of a triangle.
Distance between points A and B:
AB = √[(5-2)2 + (7-3)2]
=> AB = √9+16= √25
AB = 5 unit
Distance between points B and C:
BC = √[(8-5)2 + (1-7)2]
=> BC = √[9+36] = √45
BC = 6.708 unit
Distance between points C and A:
CA = √[(8-2)2 + (1-3)2]
=> CA = √[36+4] = √40
CA = 6.325 unit
Therefore, the length of the longest side of triangle is 6.708 unit.
Practice Problems on Euclidean Distance
P1: Calculate the Euclidean distance between points P(1, 8, 3) and Q(6, 6, 8).
P2: A car travels from point A(0, 0) to point B(5, 12). Calculate the distance traveled by the car?
P3: An airplane flies from point P(0, 0, 0) to point Q(100, 200, 300). Calculate the distance traveled by the airplane.
P4: A triangle has vertices at points M(1, 2), N(4, 6), and O(7, 3). Find the perimeter of the triangle.
P5: On a graph with points K(2, 3) and L(5, 7), plot these points and calculate the Euclidean distance between them.
P6: A drone needs to fly from point A(1, 1) to point B(10, 10). Find the shortest path the drone should take to conserve battery?
P7: A robotic arm moves from position J(1, 2, 3) to position K(4, 5, 6). Calculate the total distance traveled by the robotic arm.
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Summary – Euclidean Distance
Euclidean Distance is a metric for measuring the distance between two points in Euclidean space, reflecting the length of the shortest path connecting them, which is a straight line. The formula for calculating Euclidean Distance depends on the dimensionality of the space. In a 2-dimensional plane, the distance d between points is, d = d = √[(x2 – x1)2 + (y2 – y1)2]. In 3D, d = √[(x2 – x1)2 + (y2 – y1)2+ (z2 – z1)2].
FAQs on Euclidean Distance
Define Euclidean Distance.
Euclidean distance measures the straight-line distance between two points in Euclidean space.
What is the distance formula for a 2D Euclidean Space?
Euclidean Distance between two points (x1, y1) and (x2, y2) in using the formula:
d = √[(x2 – x1)2 + (y2 – y1)2]
What are some properties of Euclidean Distance?
- Euclidean distance is always non-negative because it represents a physical distance in space, which cannot be a negative value.
- Distance between a point and itself is always zero
Can Euclidean Distance be negative?
Euclidean Distance can’t be negative as it represents a physical distance. It can be a zero value but can’t be a negative value.
How can Euclidean Distance be extended to higher dimensions?
In general, the Euclidean Distance formula between two points (x11, x12, x13, …., x1n) and (x21, x22, x23, …., x2n) in an n-dimensional space is given by the formula:
d = √[∑(x2i – x1i)2]
What is the difference between Euclidean Distance and Manhattan Distance?
Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space;
Euclidean Distance between them is given by using the formula:
d = √[(x2 – x1)2 + (y2 – y1)2], (Calculates the square root of the sum of squared differences)
Manhattan Distance between them is given by using the formula:
d = [|x2 – x1| + |y2 – y1|], (Calculates the distance between two points as the sum of the absolute differences in their coordinates)
What is the significance of Euclidean Distance in machine learning and data analysis?
Euclidean distance is helpful in many machine learning algorithms, such as clustering, for grouping similar data points and classification techniques like K-NN. It also plays a vital role in dimensionality reduction, which helps in understanding high-dimensional data.
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